Proof by Contradiction

Concept Explained: Proof by Contradiction

  • Proof by Contradiction, also known as Reductio ad absurdum, is a method of demonstrating that a statement is true by showing that it would lead to a contradiction if it were false.

  • This form of proof is useful when it is simpler to assume the opposite of the statement and then arrive at an impossible or absurd consequence.

Using Proof by Contradiction

  • To use Proof by Contradiction, assume that the proposition you wish to prove is false.

  • This assumption is added to your existing set of known truths and you then explore the logical consequences.

  • You are searching for a logical contradiction, which can be defined as a result or statement that conflicts with a previously established fact.

  • Finding this contradiction means that the assumption (that the proposition is false) must be wrong.

  • Therefore, if assuming the statement is false leads to a contradiction, the original statement must be true.

Example of Proof by Contradiction

  • As a direct illustration, let’s prove that there is no smallest positive rational number.

  • Assume towards contradiction, that there is a smallest positive rational number. Let’s call it ‘a’.

  • Then ‘a/2’ must be smaller than ‘a’, which contradicts the assumption that ‘a’ is the smallest positive rational number.

  • So, our assumption is contradicted and we conclude that there cannot be a smallest positive rational number.

Impact of Proof by Contradiction

  • A Proof by Contradiction might not give you a direct construction or an explanatory reason why something is the case.

  • This kind of proof can be counterintuitive, as you assume the opposite of what you want to prove.

  • Remember that Proof by Contradiction is widely accepted and used in all branches of mathematics as a powerful proof technique.

Tips for Constructing a Proof by Contradiction

  • Keep a clear path of logic and don’t lose sight of what you’re trying to prove. It’s easy to get lost while exploring the consequence of the assumption.

  • Be open to finding contradictions in unexpected places. Your contradiction could involve concepts or facts outside the immediate proposition.

  • Be precise and clear in identifying and stating your contradiction. Proofs rely on solid logical steps, so ensure your contradiction is unequivocally contradictory.

  • Don’t forget to conclude your proof by stating the original proposition has been proven true, because the assumption that it was false led to a contradiction.